A060311 -id:A060311 - cp's OEIS frontend

A324162 Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 0, 1, 0, 2, 1, 0, 5, 3, 1, 0, 15, 10, 6, 1, 0, 52, 45, 25, 10, 1, 0, 203, 241, 100, 65, 15, 1, 0, 877, 1428, 511, 350, 140, 21, 1, 0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1, 0, 21147, 67035, 29765, 9030, 6951, 2646, 462, 36, 1, 0, 115975, 524926, 250200, 60355, 42651, 22827, 5880, 750, 45, 1

Offset: 0

Alois P. Heinz, Sep 02 2019

			T(4,2) = 10: 123/4, 124/3, 12/34, 134/2, 13/24, 14/23, 1/234, 1/2|3/4, 1/3|2/4, 1/4|2/3.
Triangle T(n,k) begins:
  1;
  0,    1;
  0,    2,    1;
  0,    5,    3,    1;
  0,   15,   10,    6,    1;
  0,   52,   45,   25,   10,    1;
  0,  203,  241,  100,   65,   15,   1;
  0,  877, 1428,  511,  350,  140,  21,  1;
  0, 4140, 9325, 3626, 1736, 1050, 266, 28, 1;
  ...

Alois P. Heinz, Rows n = 0..140, flattened
Wikipedia, Partition of a set

Columns k=0-10 give: A000007, A000110 (for n>0), A060311, A327504, A327505, A327506, A327507, A327508, A327509, A327510, A327511.
Row sums give A324238.
T(2n,n) gives A324241.
Cf. A008277, A048993, A039810, A130191, A325929.

T:= proc(n, k) option remember; `if`(n=0, 1, `if`(k=0, 0, add(
      T(n-j, k)*binomial(n-1, j-1)*Stirling2(j, k), j=k..n)))
    end:
seq(seq(T(n, k), k=0..n), n=0..12);

nmax = 10;
col[k_] := col[k] = CoefficientList[Exp[(Exp[x]-1)^k/k!] + O[x]^(nmax+1), x][[k+1;;]] Range[k, nmax]!;
T[n_, k_] := Which[k == n, 1, k == 0, 0, True, col[k][[n-k+1]]];
Table[T[n, k], {n, 0, nmax}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 26 2020 *)

T(n, k) = if(k==0, 0^n, sum(j=0, n\k, (k*j)!*stirling(n, k*j, 2)/(k!^j*j!))); \\ Seiichi Manyama, May 07 2022

E.g.f. of column k>0: exp((exp(x)-1)^k/k!).
Sum_{k=1..n} k * T(n,k) = A325929(n).
T(n,k) = Sum_{j=0..floor(n/k)} (k*j)! * Stirling2(n,k*j)/(k!^j * j!) for k > 0. - Seiichi Manyama, May 07 2022

A052859 Expansion of e.g.f.: exp(exp(2x) - 2exp(x) + 1).

Original entry on oeis.org

1, 0, 2, 6, 26, 150, 962, 6846, 54266, 471750, 4439762, 44911086, 485570186, 5581383990, 67890295202, 870493380126, 11726471352986, 165475293394470, 2439632685738482, 37491028556508366, 599285435979866666, 9945441791592272790, 171062503783616702402

Offset: 0

encyclopedia(AT)pommard.inria.fr, Jan 25 2000

Previous name was: A simple grammar.

a(n) is the number of ways to select a nonempty proper subset from each block of the set partitions of {1,2,...,n}. - Geoffrey Critzer, Jan 20 2012

Vincenzo Librandi, Table of n, a(n) for n = 0..200
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 827
Vaclav Kotesovec, Asymptotics for a certain group of exponential generating functions, arXiv:2207.10568 [math.CO], Jul 13 2022 (set m=1, b=2, r=-2, d=1, s=1).
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)

Cf. A060311, A067994.

spec := [S,{B=Prod(C,C),C=Set(Z,1 <= card),S=Set(B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
# second Maple program:
a:= proc(n) option remember; `if`(n=0, 1, add(a(n-j)
      *2*binomial(n-1, j-1)*Stirling2(j, 2), j=1..n))
    end:
seq(a(n), n=0..25);  # Alois P. Heinz, Sep 02 2019

nn=20; a=Exp[x]-1; Range[0,nn]! CoefficientList[Series[Exp[a^2], {x,0,nn}], x]  (* Geoffrey Critzer, Jan 20 2012 *)
Table[Sum[BellY[n, k, 2^Range[n] - 2], {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Nov 09 2016 *)
Table[Sum[(2*k)!*StirlingS2[n, 2*k]/k!, {k, 0, n/2}], {n, 0, 25}] (* Vaclav Kotesovec, Oct 04 2022 *)

my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (2*k)!*x^(2*k)/(k!*prod(j=1, 2*k, 1-j*x)))) \\ Seiichi Manyama, May 07 2022

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/k!); \\ Seiichi Manyama, May 07 2022

E.g.f.: exp(exp(x)^2-2*exp(x)+1).
Stirling transform of unsigned Hermite numbers: Sum_{k=0..n} Stirling2(n, k)*|HermiteH(k, 0)|. - Vladeta Jovovic, Sep 12 2003
From Seiichi Manyama, May 07 2022: (Start)
G.f.: Sum_{k>=0} (2*k)! * x^(2*k)/(k! * Product_{j=1..2*k} (1 - j * x)).
a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/k!. (End)
a(n) ~ 2^n * exp(1/2 - n - 2*sqrt(n/LambertW(n)) + n/LambertW(n)) * n^n / (sqrt(1 + LambertW(n)) * LambertW(n)^n). - Vaclav Kotesovec, Oct 04 2022

New name using e.g.f. from Vaclav Kotesovec, Oct 04 2022

A347001 Expansion of e.g.f. exp( log(1 - x)^2 / 2 ).

Original entry on oeis.org

1, 0, 1, 3, 14, 80, 544, 4284, 38310, 383256, 4239006, 51345690, 675770028, 9600349824, 146396925648, 2384700728760, 41320373582652, 758780222426592, 14718569154071964, 300706641183038292, 6453691377726073128, 145154958710291611200, 3414131149418742544320

Offset: 0

Ilya Gutkovskiy, Aug 10 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..448

Cf. A000254, A060311, A305306, A346921, A347002, A347003, A347004.

nmax = 22; CoefficientList[Series[Exp[Log[1 - x]^2/2], {x, 0, nmax}], x] Range[0, nmax]!
a[0] = 1; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1] Abs[StirlingS1[k, 2]] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 22}]

a(n) = sum(k=0, n\2, (2*k)!*abs(stirling(n, 2*k, 1))/(2^k*k!)); \\ Seiichi Manyama, May 06 2022

a(0) = 1; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * |Stirling1(k,2)| * a(n-k).

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * |Stirling1(n,2*k)|/(2^k * k!). - Seiichi Manyama, May 06 2022

A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

Original entry on oeis.org

1, 0, 0, 6, 12, 20, 390, 2562, 11816, 105912, 1063530, 8815070, 81342492, 895185876, 9971185406, 112642410090, 1372455608400, 17750397057392, 236950003516626, 3286258330135734, 47688868443593540, 719345273005797900, 11222288509573985382, 181168865439054099266

Offset: 0

Vaclav Kotesovec, Aug 06 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 0..200
Vaclav Kotesovec, Asymptotic solution of the equations using the Lambert W-function

Column k=2 of A292892.
Cf. A000110, A052506, A216507, A060311.
Cf. A216688, A216689, A080108, A000248.

CoefficientList[Series[E^(x^2*(E^x-1)), {x, 0, 20}], x] * Range[0, 20]!

x='x+O('x^30); Vec(serlaplace(exp(x^2*(exp(x) - 1)))) \\ G. C. Greubel, Nov 21 2017

a(n) = n!*sum(k=0, n\3, stirling(n-2*k, k, 2)/(n-2*k)!); \\ Seiichi Manyama, Jul 09 2022

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=(i-1)!*sum(j=3, i, j/(j-2)!*v[i-j+1]/(i-j)!)); v; \\ Seiichi Manyama, Jul 09 2022

a(n) ~ exp((n-r^3)/(2+r)-n) * n^(n+1/2) / (r^n * sqrt((2*r^3*(3+r) + n*(1+r)*(4+r))/(2+r))), where r is the root of the equation r^2*((2+r) * exp(r) - 2) = n.
(a(n)/n!)^(1/n) ~ exp(1/(3*LambertW(n^(1/3)/3))) / (3*LambertW(n^(1/3)/3)).
From Seiichi Manyama, Jul 09 2022: (Start)
a(n) = n! * Sum_{k=0..floor(n/3)} Stirling2(n-2*k,k)/(n-2*k)!.
a(0) = 1; a(n) = (n-1)! * Sum_{k=3..n} k/(k-2)! * a(n-k)/(n-k)!. (End)

A346974 Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).

Original entry on oeis.org

1, 3, 4, -15, -134, -357, 2374, 33645, 133186, -1288617, -24887906, -130115895, 1666879306, 40612637523, 262868197414, -4221449488635, -123802488449774, -952293015617937, 18497401668708334, 632675912865355425, 5622243546094977946, -128799294291220310997

Offset: 2

Ilya Gutkovskiy, Aug 09 2021

sign

Cf. A000225, A003704, A060311, A346975, A346976, A346977.

nmax = 23; CoefficientList[Series[Log[1 + (Exp[x] - 1)^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Drop[#, 2] &
a[n_] := a[n] = StirlingS2[n, 2] - (1/n) Sum[Binomial[n, k] StirlingS2[n - k, 2] k a[k], {k, 1, n - 1}]; Table[a[n], {n, 2, 23}]

a(n) = Stirling2(n,2) - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Stirling2(n-k,2) * k * a(k).
a(n) ~ -n! * 2^(n+1) * cos(n*arctan(2*arctan(sqrt(2))/log(3))) / (n * (4*arctan(sqrt(2))^2 + log(3)^2)^(n/2)). - Vaclav Kotesovec, Aug 09 2021
a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1) * (2*k)! * Stirling2(n,2*k)/(k * 2^k). - Seiichi Manyama, Jan 23 2025

A354395 Expansion of e.g.f. exp( -(exp(x) - 1)^2 / 2 ).

Original entry on oeis.org

1, 0, -1, -3, -4, 15, 149, 672, 1091, -12855, -162796, -1060653, -2925319, 30881760, 598929239, 5688937797, 29126981516, -112222099065, -4930674413971, -69798552313728, -598032658869829, -1296500625378255, 65193402297999524, 1515140106814565547

Offset: 0

Seiichi Manyama, May 25 2022

sign

Cf. A000587, A354396, A354397, A354398.

Cf. A060311, A354391.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp(-(exp(x)-1)^2/2)))

a_vector(n) = my(v=vector(n+1)); v[1]=1; for(i=1, n, v[i+1]=-sum(j=1, i, binomial(i-1, j-1)*stirling(j, 2, 2)*v[i-j+1])); v;

a(n) = sum(k=0, n\2, (2*k)!*stirling(n, 2*k, 2)/((-2)^k*k!));

a(0) = 1; a(n) = -Sum_{k=1..n} binomial(n-1,k-1) * Stirling2(k,2) * a(n-k).

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * Stirling2(n,2*k)/((-2)^k * k!).

A353894 Expansion of e.g.f. exp( (x * (exp(x) - 1))^2 / 4 ).

Original entry on oeis.org

1, 0, 0, 0, 6, 30, 105, 315, 2128, 24948, 251415, 2093025, 16437036, 148728294, 1693067467, 21459867975, 270217289280, 3338860150488, 42428729660751, 581966068060485, 8654787480759700, 135253842794286930, 2163416823356628147, 35313421249845594075

Offset: 0

Seiichi Manyama, May 09 2022

nonn

Cf. A052506, A353895, A353896.

Cf. A060311, A353883, A353891.

my(N=30, x='x+O('x^N)); Vec(serlaplace(exp((x*(exp(x)-1))^2/4)))

a(n) = n!*sum(k=0, n\4, (2*k)!*stirling(n-2*k, 2*k, 2)/(4^k*k!*(n-2*k)!));

a(n) = n! * Sum_{k=0..floor(n/4)} (2*k)! * Stirling2(n-2*k,2*k)/(4^k * k! * (n-2*k)!).

A357031 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^2 / 2.

Original entry on oeis.org

1, 0, 1, 3, 22, 195, 2131, 28623, 445789, 7982355, 161208976, 3626200743, 89942239861, 2438520508515, 71754865476841, 2277574224716703, 77570723071721938, 2821841221403098995, 109200125293424385271, 4479379126010806153143, 194148151869063307919725

Offset: 0

Seiichi Manyama, Sep 09 2022

nonn

Cf. A030019, A357032.

Cf. A060311, A357024.

m = 21; (* number of terms *)
A[_] = 0;
Do[A[x_] = Exp[(Exp[x*A[x]] - 1)^2/2] + O[x]^m // Normal, {m}];
CoefficientList[A[x], x]*Range[0, m - 1]! (* Jean-François Alcover, Sep 12 2022 *)

a(n) = sum(k=0, n\2, (2*k)!*(n+1)^(k-1)*stirling(n, 2*k, 2)/(2^k*k!));

a(n) = Sum_{k=0..floor(n/2)} (2*k)! * (n+1)^(k-1) * Stirling2(n,2*k)/(2^k * k!).

cp's OEIS Frontend

A324162 Number T(n,k) of set partitions of [n] where each subset is again partitioned into k nonempty subsets; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

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A052859 Expansion of e.g.f.: exp(exp(2*x) - 2*exp(x) + 1).

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Extensions

A347001 Expansion of e.g.f. exp( log(1 - x)^2 / 2 ).

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Mathematica

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A240989 Expansion of e.g.f. exp(x^2 * (exp(x) - 1)).

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Mathematica

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A346974 Expansion of e.g.f. log( 1 + (exp(x) - 1)^2 / 2 ).

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Mathematica

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A354395 Expansion of e.g.f. exp( -(exp(x) - 1)^2 / 2 ).

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PARI

PARI

PARI

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A353894 Expansion of e.g.f. exp( (x * (exp(x) - 1))^2 / 4 ).

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PARI

PARI

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A357031 E.g.f. satisfies log(A(x)) = (exp(x * A(x)) - 1)^2 / 2.

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A052859 Expansion of e.g.f.: exp(exp(2x) - 2exp(x) + 1).